# Relativistic addition of velocities problem

## Homework Statement

You are driving down a two-lane highway and a truck in the opposite lane travels toward you. Suppose the speed of light in a vacuum is 65 m/s. Determine the speed of the truck relative to you when

a - your speed is 25 m/s and the truck's speed is 35 m/s and

b - your speed is 5 m/s and the truck's speed is 55 m/s

The speeds given are relative to the ground

## Homework Equations

V(ab) = Vac + Vcb / 1 + (Vac*Vcb/c^2)

## The Attempt at a Solution

The velocity of the car (me) relative to the ground is Vac or 25 or 0.25c
Velocity of truck in opposite direction is Vcb or -35 or -0.35c

Plugging in to relativistic addition of velocities:

0.25c - 0.35c / 1 - (0.0.875)= -0.12c
Given that c = 65 for this problem, Vab = (-0.12)(65) = -7.1 m/s

I'm pretty sure that answer is wrong, because it makes no sense that the truck would be traveling at 7m/s relative to the car but I cannot figure out what I'm doing wrong here.

And for part B, I'm a little unclear on how "close" to the speed of light the velocities have to be in order to use the relativistic equation. Is 5 m/s far enough from the given speed of light (65 m/s) that I should use the "original recipe" addition of velocities to find the velocity of the truck? How would I know this?

Any help is greatly appreciated!

Doc Al
Mentor
The velocity of the car (me) relative to the ground is Vac or 25 or 0.25c
Va/c = 25 m/s (not 0.25c); (Va/c is the speed of you "a" relative to the ground "c").

Velocity of truck in opposite direction is Vcb or -35 or -0.35c
Careful, Vc/b is the velocity of the ground with respect to the truck ("b"), thus Vc/b = + 35 m/s.

The speed of light is 65 m/s. Va/c = 25 m/s ≠ .25c

Oh! So, Vac = 25/65 = 0.38c and Vcb = 35/65 = 0.58c

So, plugging in:

0.38c+0.54c / 1 + (0.38c)(0.54c)/c^2
= 0.71c, so truck is travelling 49m/s relative to the car? That makes more sense, because it would seem to be travelling faster than it actually is when viewed from the car's frame. Is that right?

For Part B, can I use the relativistic equation or is 5 m/s not close enough to the speed of light?

Doc Al
Mentor
Oh! So, Vac = 25/65 = 0.38c and Vcb = 35/65 = 0.58c

So, plugging in:

0.38c+0.54c / 1 + (0.38c)(0.54c)/c^2
= 0.71c, so truck is travelling 49m/s relative to the car? That makes more sense, because it would seem to be travelling faster than it actually is when viewed from the car's frame. Is that right?
Yes. The speed of the truck in the frame of the car is greater than the speed of the truck in the frame of the earth (since the car is also moving with respect to the earth).

For Part B, can I use the relativistic equation or is 5 m/s not close enough to the speed of light?
Unless both speeds are much smaller than the speed of light, it's best to use the relativistic formula. (For small speeds, the relativistic formula gives almost the same results as the galilean formula.)

OK, I think I understand a little more. Thanks so much for helping me!